Mathematische Annalen

, Volume 355, Issue 2, pp 783–799 | Cite as

A simple separable exact C*-algebra not anti-isomorphic to itself

  • N. Christopher Phillips
  • Maria Grazia ViolaEmail author


We give an example of an exact, stably finite, simple, separable C*-algebra D which is not isomorphic to its opposite algebra. Moreover, D has the following additional properties. It is stably finite, approximately divisible, has real rank zero and stable rank one, has a unique tracial state, and the order on projections over D is determined by traces. It also absorbs the Jiang-Su algebra Z, and in fact absorbs the 3 UHF algebra. We can also explicitly compute the K-theory of D, namely \({K_0 (D) \cong {\mathbb{Z}} [ \tfrac{1}{3}]}\) with the standard order, and K 1 (D) =  0, as well as the Cuntz semigroup of D, namely \({W (D) \cong {\mathbb{Z}} [ \tfrac{1}{3} ]_{+} \sqcup (0, \infty).}\)

Mathematics Subject Classification (2000)

46L35 46L37 46L40 46L54 


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  1. 1.
    Avitzour D.: Free products of C*-algebras. Trans. Am. Math. Soc. 271, 423–435 (1982)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Blackadar, B.: Comparison theory for simple C*-algebras. In: Evans, D.E., Takesaki M. (eds.) Operator Algebras and Applications, pp. 21–54. Lecture Notes Series, vol. 135. London Mathematical Society, Cambridge University Press, Cambridge, New York (1988)Google Scholar
  3. 3.
    Blackadar B., Handelman D.: Dimension functions and traces on C*-algebras. J. Funct. Anal. 45, 297–340 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Blackadar B., Kumjian A., Rørdam M.: Approximately central matrix units and the structure of non-commutative tori. K-Theory 6, 267–284 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Blackadar B., Rørdam M.: Extending states on preordered semigroups and the existence of quasitraces on C*-algebras. J. Algebra 152, 240–247 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Brown N.P., Perera F., Toms A.S.: The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras. J. Reine Angew. Math. 621, 191–211 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Connes A.: Sur la classification des facteurs de typeII. C. R. Acad. Sci. Paris Sér.A 281, 13–15 (1975)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Connes A.: A factor not anti-isomorphic to itself. Ann. Math. 101(2), 536–554 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Connes A.: Periodic automorphisms of the hyperfinite factor of type. Acta Sci. Math. (Szeged) 39, 39–66 (1977)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cuntz J., Pedersen G.K.: Equivalence and traces on C*-algebras. J. Funct. Anal. 33, 135–164 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dykema K.: Exactness of reduced amalgamated free product C*-algebras. Forum Math. 16, 161–180 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Germain E.: KK-theory of reduced free product C*-algebras. Duke Math. J 82, 707–723 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Haagerup, U.: Quasitraces on exact C*-algebras are traces (handwritten manuscript, 1991)Google Scholar
  14. 14.
    Haagerup U., Thorbjørnsen S.: Random matrices and K-theory for exact C*-algebras. Documenta Math 4, 341–450 (1999) (electronic)zbMATHGoogle Scholar
  15. 15.
    Hirshberg I., Winter W.: Rokhlin actions and self-absorbing C*-algebras. Pacific J. Math 233, 125–143 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Izumi M.: Finite group actions on C*-algebras with the Rohlin property. I. Duke Math. J. 122, 233–280 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Jiang X., Su H.: On a simple unital projectionless C*-algebra. Am. J. Math. 121, 359–413 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Kirchberg E.: On nonsemisplit extensions, tensor products and exactness of group C*-algebras. Invent. Math. 112, 449–489 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Kirchberg E.: On subalgebras of the CAR algebra. J. Funct. Anal. 129, 35–63 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Kirchberg E.: Commutants of unitaries in UHF algebras and functorial properties of exactness. J. Reine Angew. Math. 452, 39–77 (1994)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kishimoto A.: Outer automorphisms and reduced crossed products of simple C*-algebras. Commun. Math. Phys. 81, 429–435 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Osaka, H., Phillips, N.C.: Crossed products by finite group actions with the Rokhlin property. Math. Z., to appearGoogle Scholar
  23. 23.
    Phillips N.C.: A simple separable C*-algebra not isomorphic to its opposite algebra. Proc. Am. Math. Soc. 132, 2997–3005 (2004)zbMATHCrossRefGoogle Scholar
  24. 24.
    Rørdam M.: On the structure of simple C*-algebras tensored with a UHF-algebra. J. Funct. Anal. 100, 1–17 (1991)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rørdam M.: On the structure of simple C*-algebras tensored with a UHF-algebra. II. J. Funct. Anal. 107, 255–269 (1992)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rosenberg J.: Appendix to O. Bratteli’s paper on Crossed products of UHF algebras. Duke Math. J. 46, 25–26 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Schochet C.: Topological methods for C*-algebras II: geometric resolutions and the Künneth formula. Pacific J. Math. 98, 443–458 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Skandalis G.: Kasparov’s bivariant K-theory and applications. Expos. Math. 9, 193–250 (1991)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Strătilă, Ş., Zsidó, L.: Lectures on von Neumann algebras, Chapter A. translated from the Romanian by S. Teleman, Editura Academiei, Bucharest and Abacus Press, Tunbridge Wells (1979)Google Scholar
  30. 30.
    Viola M.G.: On a subfactor construction of a factor non-antiisomorphic to itself. Int. J. Math. 15, 833–854 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Wassermann S.: On tensor products of certain group C*-algebras. J. Funct. Anal. 23, 239–254 (1976)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OregonEugeneUSA
  2. 2.Department of Mathematics and Interdisciplinary StudiesLakehead University-OrilliaOrilliaCanada

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